We consider the problem of sampling transition paths between two given metastable states of a molecular system, eg. a folded and unfolded protein or products and reactants of a chemical reaction. Due to the existence of high energy barriers separating the states, these transition paths are unlikely to be sampled with standard Molecular Dynamics (MD) simulation. Traditional methods to augment MD with a bias potential to increase the probability of the transition rely on a dimensionality reduction step based on Collective Variables (CVs). Unfortunately, selecting appropriate CVs requires chemical intuition and traditional methods are therefore not always applicable to larger systems. Additionally, when incorrect CVs are used, the bias potential might not be minimal and bias the system along dimensions irrelevant to the transition. Showing a formal relation between the problem of sampling molecular transition paths, the Schrodinger bridge problem and stochastic optimal control with neural network policies, we propose a machine learning method for sampling said transitions. Unlike previous non-machine learning approaches our method, named PIPS, does not depend on CVs. We show that our method successful generates low energy transitions for Alanine Dipeptide as well as the larger Polyproline and Chignolin proteins.

We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. Equivariant SVGD explicitly incorporates symmetry information in a density through equivariant kernels which makes the resultant sampler efficient both in terms of sample complexity and the quality of generated samples. Subsequently, we define equivariant energy based models to model invariant densities that are learned using contrastive divergence. By utilizing our equivariant SVGD for training equivariant EBMs, we propose new ways of improving and scaling up training of energy based models. We apply these equivariant energy models for modelling joint densities in regression and classification tasks for image datasets, many-body particle systems and molecular structure generation.

Methods for adversarial robustness certification aim to provide an upper bound on the test error of a classifier under adversarial manipulation of its input. Current certification methods are computationally expensive and limited to attacks that optimize the manipulation with respect to a norm. We overcome these limitations by investigating the robustness properties of Nearest Prototype Classifiers (NPCs) like learning vector quantization and large margin nearest neighbor. For this purpose, we study the hypothesis margin. We prove that if NPCs use a dissimilarity measure induced by a seminorm, the hypothesis margin is a tight lower bound on the size of adversarial attacks and can be calculated in constant time—this provides the first adversarial robustness certificate calculable in reasonable time. Finally, we show that each NPC trained by a triplet loss maximizes the hypothesis margin and is therefore optimized for adversarial robustness. In the presented evaluation, we demonstrate that NPCs optimized for adversarial robustness are competitive with state-of-the-art methods and set a new benchmark with respect to computational complexity for robustness certification.

Abstract Neural networks are state-of-the-art classification approaches but are generally difficult to interpret. This issue can be partly alleviated by constructing a precise decision process within the neural network. In this work, a network architecture, denoted as Classification-By-Components network (CBC), is proposed. It is restricted to follow an intuitive reasoning based decision process inspired by Biederman's recognition-by-components theory from cognitive psychology. The network is trained to learn and detect generic components that characterize objects. In parallel, a class-wise reasoning strategy based on these components is learned to solve the classification problem. In contrast to other work on reasoning, we propose three different types of reasoning: positive, negative, and indefinite. These three types together form a probability space to provide a probabilistic classifier. The decomposition of objects into generic components combined with the probabilistic reasoning provides by design a clear interpretation of the classification decision process. The evaluation of the approach on MNIST shows that CBCs are viable classifiers. Additionally, we demonstrate that the inherent interpretability offers a profound understanding of the classification behavior such that we can explain the success of an adversarial attack. The method's scalability is successfully tested using the ImageNet dataset.